How to make an original image texture with transparency in Blender Render?
Okay, it would appear I've managed fix the transparency issue on my own with the aid of this video tutorial enter link description here that was included on my YouTube Recommendations this morning.  While following along with the video, I discovered that the reason for the faded look of the pupil texture was that the Diffuse Influence of the Alpha wasn't enabled in the Texture tab.  Once the Alpha was enabled, the pupil looked the way that it was supposed to.
As for the eyelid(s) texture, I didn't actually find out what was wrong with it.  Instead, I created a completely blank image texture in Gimp and imported onto my model.  I then repainted the eyelid(s) onto the new blank image texture, and as a result, the new eyelid texture now has looks the way that is was intended without any apparent issues that were found in the old version.
However, rendering an image still results in the same problem as before, with a weird visible mesh around the eyes that only appears when a image is rendered and not in any other mode in Blender.  I have no idea if this a serious issue or not, I guess I would have to import the model &amp; its textures into Unreal Engine 4 to see if there is any visible problems whatsoever.